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1、平台式惯导系统的局限性
费用高，结构复杂，故障率较高，可靠性差
2、定义
将惯性器件（陀螺仪和加速度计）直接安装在载体上的一种惯导系统
3、与平台式惯导系统的区别
去掉了实体的惯性平台而代之以存贮在计算机里的**“数学..." />
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1、平台式惯导系统的局限性
费用高，结构复杂，故障率较高，可靠性差
2、定义
将惯性器件（陀螺仪和加速度计）直接安装在载体上的一种惯导系统
3、与平台式惯导系统的区别
去掉了实体的惯性平台而代之以存贮在计算机里的**“数学...">
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1、平台式惯导系统的局限性
费用高，结构复杂，故障率较高，可靠性差
2、定义
将惯性器件（陀螺仪和加速度计）直接安装在载体上的一种惯导系统
3、与平台式惯导系统的区别
去掉了实体的惯性平台而代之以存贮在计算机里的**“数学...">
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                                <h2>
                                    5.1 捷联惯导系统
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                                    2023-12-11, 659 words, 3 min read
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                                        <h2 id="一-捷联惯导系统">一、捷联惯导系统</h2>
<h4 id="1-平台式惯导系统的局限性">1、平台式惯导系统的局限性</h4>
<p>费用高，结构复杂，故障率较高，可靠性差</p>
<h4 id="2-定义">2、定义</h4>
<p>将惯性器件（陀螺仪和加速度计）直接安装在载体上的一种惯导系统</p>
<h4 id="3-与平台式惯导系统的区别">3、与平台式惯导系统的区别</h4>
<p>去掉了实体的<strong>惯性平台</strong>而代之以存贮在计算机里的**“数学平台”**</p>
<h2 id="二-捷联式惯导系统的工作原理">二、捷联式惯导系统的工作原理</h2>
<p>1、三个加速度计测得的比力是沿载体坐标系 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>o</mi><msub><mi>x</mi><mi>b</mi></msub><msub><mi>y</mi><mi>b</mi></msub><msub><mi>z</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">ox_by_bz_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">o</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的比力分量<img src="http://cos.pansis.site/202312071917630.png/abc123" alt="image-20231207191702538" style="zoom:43%;" /></p>
<p>实际中需要的是沿导航坐标系n的比力分量 <img src="http://cos.pansis.site/202312071918215.png/abc123" alt="image-20231207191823178" style="zoom:53%;" /></p>
<p>因此需要进行转换，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mi>n</mi></msup><mo>=</mo><msubsup><mi>C</mi><mi>b</mi><mi>n</mi></msubsup><msup><mi>f</mi><mi>b</mi></msup></mrow><annotation encoding="application/x-tex">f^n=C^n_bf^b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4168920000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span></span></span></span></span></span></span></span></p>
<p>2、要求解随时间变化着的姿态矩阵 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>C</mi><mi>b</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">C^n_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9664379999999999em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4168920000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span></span></span></span>，须求解微分方程组<img src="http://cos.pansis.site/202312071921058.png/abc123" alt="image-20231207192112012" style="zoom:50%;" /></p>
<p>其中，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi mathvariant="normal">Ω</mi><mrow><mi>n</mi><mi>b</mi></mrow><mi>b</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega ^b_{nb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">b</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span></span></span></span>为姿态角速度 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>w</mi><mrow><mi>n</mi><mi>b</mi></mrow><mi>b</mi></msubsup></mrow><annotation encoding="application/x-tex">w ^b_{nb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-2.4168920000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">b</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span></span></span></span> 的反对称矩阵，<img src="http://cos.pansis.site/202312071926103.png/abc123" alt="image-20231207192630048" style="zoom:43%;" /></p>
<p>3、下面求解姿态角速度 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>w</mi><mrow><mi>n</mi><mi>b</mi></mrow><mi>b</mi></msubsup></mrow><annotation encoding="application/x-tex">w ^b_{nb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-2.4168920000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">b</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>$w<sup>b_{nb}=w</sup>b_{ib}-w<sup>b_{in}=w</sup>b_{ib}-C_n<sup>bw_{in}</sup>n=w<sup>b_{ib}-C_n</sup>b(w_{ie}<sup>n+w_{en}</sup>n) $</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>w</mi><mrow><mi>i</mi><mi>b</mi></mrow><mi>b</mi></msubsup></mrow><annotation encoding="application/x-tex">w^b_{ib}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-2.4168920000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">b</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span></span></span></span> 由固定在载体上的三个陀螺仪测得，它们的测量轴也分别与载体坐标系b的三个轴相重合</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>w</mi><mrow><mi>i</mi><mi>e</mi></mrow><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">w^n_{ie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.923056em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">e</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> <img src="http://cos.pansis.site/202312071935412.png/abc123" alt="image-20231207193552355" style="zoom:33%;" /></p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>w</mi><mrow><mi>e</mi><mi>n</mi></mrow><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">w^n_{en}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.911392em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> <img src="http://cos.pansis.site/202312071936743.png/abc123" alt="image-20231207193640681" style="zoom:33%;" /></p>
<p>4、原理总结</p>
<ul>
<li>正是依靠计算机对姿态矩阵的计算提供出<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>C</mi><mi>b</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">C_b^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9664379999999999em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4168920000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span></span></span></span>的即时值，才能顺利的由<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mi>b</mi></msup></mrow><annotation encoding="application/x-tex">f^b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.043548em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span></span></span></span></span></span></span></span>得到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span></span></span></span> ，从而进行 有效的导航计算。</li>
<li>正是在这个意义上，用计算机软件实现了一个“数学平台”，并取代了原有的硬件 （实体）平台</li>
<li>在姿态矩阵计算之前，也必须提供初始矩阵<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>C</mi><mi>b</mi><mi>n</mi></msubsup><mo>(</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">C^n_b(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.033108em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4168920000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span></li>
</ul>
<h2 id="三-捷联式惯导系统特点">三、捷联式惯导系统特点</h2>
<h4 id="1-特点">1、特点</h4>
<ul>
<li>
<p>取消了实体平台，代之以大量的实时软件，结果大大降低了系统体积、重量和成本；</p>
</li>
<li>
<p>减少了系统中的机电元件，对加速度计和陀螺仪容 易实现多余度配置方案，因此工作的可靠性大大提高；</p>
</li>
<li>
<p>较平台系统维护简便，故障率低；体积小、成本低、对准速度快</p>
</li>
<li>
<p>由于动态环境恶劣，故对惯性器件的要求比平台系统高。</p>
<ul>
<li>
<p>要求陀螺仪能测量小至 0.01°/h， 大至400°/s的转动角速度，其动态量程达<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">10^8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span>，原因：</p>
</li>
<li>
<p>由于实体平台对惯性器件所起的运动隔离作 用已不复存在惯性器件将不得不在相当恶劣的 环境下工作。</p>
</li>
<li>
<p>载体的<strong>运动冲击</strong>和<strong>振动</strong>也将严重影响惯 性器件的性能</p>
</li>
</ul>
</li>
</ul>
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